Question: Find the smallest positive angle $\theta,$ in degrees, for which
\[\cos \theta = \sin 60^\circ + \cos 42^\circ - \sin 12^\circ - \cos 6^\circ.\]
Answer: We have that
\begin{align*}
\sin 60^\circ &= \cos 30^\circ, \\
\cos 42^\circ &= \cos (360^\circ - 42^\circ) = \cos 318^\circ, \\
-\sin 12^\circ &= -\cos (90^\circ - 12^\circ) = -\cos 78^\circ = \cos (180^\circ - 78^\circ) = \cos 102^\circ, \\
-\cos 6^\circ &= \cos (180^\circ - 6^\circ) = \cos 174^\circ,
\end{align*}so
\[\cos \theta = \cos 30^\circ + \cos 318^\circ + \cos 102^\circ + \cos 174^\circ.\]If we plot $(\cos t, \sin t)$ for $t = 30^\circ,$ $102^\circ,$ $174^\circ,$ $246^\circ,$ and $318^\circ,$ the five points form the vertices of a regular pentagon.

[asy]
unitsize(2 cm);

pair A, B, C, D, E, O;

A = dir(30);
B = dir(30 + 360/5);
C = dir(30 + 2*360/5);
D = dir(30 + 3*360/5);
E = dir(30 + 4*360/5);
O = (0,0);

draw((-1.2,0)--(1.2,0));
draw((0,-1.2)--(0,1.2));
draw(Circle(O,1));
draw(O--A);
draw(O--B);
draw(O--C);
draw(O--D);
draw(O--E);

label("$30^\circ$", A, A);
label("$102^\circ$", B, B);
label("$174^\circ$", C, C);
label("$246^\circ$", D, D);
label("$318^\circ$", E, E);
[/asy]

Then by symmetry, the sum of the $x$-coordinates is
\[\cos 30^\circ + \cos 102^\circ + \cos 174^\circ + \cos 246^\circ + \cos 318^\circ = 0.\]Thus,
\begin{align*}
\cos \theta &= -\cos 246^\circ \\
&= -\cos (360^\circ - 246^\circ) \\
&= -\cos 114^\circ \\
&= \cos (180^\circ - 114^\circ) \\
&= \cos 66^\circ.
\end{align*}Thus, the smallest such $\theta$ is $\boxed{66^\circ}.$